#6442713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,19)(8,23)(9,15)(10,16)(11,17)(12,18)(13,21)(14,26)(20,30)(22,29)(24,36)(25,35)(27,34)(28,33)(31,32);(2,4)(5,7)(6,13)(8,24)(9,18)(10,15)(11,21)(12,22)(14,31)(16,19)(17,29)(20,26)(23,30)(25,33)(27,35)(28,32)(34,36);(5,8,10,26,12,28,11,27)(6,14,17,23,18,34,16,33)(7,20,15,32,22,35,21,24)(9,25,19,31,13,30,29,36);(5,9)(6,15)(7,16)(8,25)(10,19)(11,29)(12,13)(14,32)(17,22)(18,21)(20,33)(23,35)(24,34)(26,31)(27,36)(28,30);(5,10,12,11)(6,16,18,17)(7,21,22,15)(8,26,28,27)(9,19,13,29)(14,33,34,23)(20,24,35,32)(25,31,30,36)

Polycyclic presentation:

2^9|c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349909);